Understanding the Relationship between Gas Pressure and Volume at Constant Temperature

Have you ever wondered how changing the pressure of a gas will affect its volume when the temperature remains constant? This is a fundamental concept in thermodynamics and can be explained using Boyles Law. Boyles Law states that at a constant temperature, the pressure of a given amount of gas is inversely proportional to its volume. This relationship is mathematically expressed as:

P1V1 P2V2

where: P1 is the initial pressure V1 is the initial volume P2 is the final pressure V2 is the final volume

Let's explore this concept by working through an example problem. Suppose we have a gas at an initial pressure of P1 and an initial volume of V1. If the pressure is increased by 5, we can calculate the new volume.

Example Problem

Given that the pressure increases by 5, the final pressure P2 can be expressed as:

P2 1.05 × P1

Applying Boyles Law:

Rearranging the Equation to Solve for V2

P1V1 P2V2

Thus,

V2 (P1V1) / (P2)

Substituting the value of P2:

V2 (P1V1) / (1.05P1)

Which simplifies to:

V2 (V1) / 1.05

Calculating the Decrease in Volume

To determine the decrease in volume:

Decrease in Volume V1 - V2

Substituting V2:

Decrease in Volume V1 - (V1 / 1.05)

Factoring out V1:

Decrease in Volume V1 [1 - (1 / 1.05)]

This simplifies to:

Decrease in Volume V1 [0.05 / 1.05]

Therefore, the volume decreases by approximately:

Decrease in Volume ≈ V1 × 0.0476
or about 4.76%

This means that when the pressure of a gas is increased by 5% at constant temperature, the volume of the gas decreases by approximately 4.76%.

Practical Considerations

This concept is particularly relevant in scenarios where gases are compressed, such as in cylinder valves, balloons, or even in everyday situations like filling up a tire. At low pressures, similar to those found in the atmosphere, the ideal gas law (PV nRT) is a close approximation. This law indicates that at constant temperature, the Volume is inversely proportional to the Pressure:

Initial PV Final (1.05P) Vfinal

This implies:

Vfinal 0.952Vinitial

Thus, the volume decreases to approximately 95.2% of its original value, which is a decrease of 4.8%.

Conclusion

Understanding the relationship between gas pressure and volume is crucial in many scientific and engineering applications. Whether you are studying thermodynamics, designing pressure systems, or simply curious about the behavior of gases, the principles outlined in Boys Law help us comprehend how changes in pressure influence gas volume.